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Sequential discretisation schemes for a class of stochastic differential equations and their application to Bayesian filtering

Deniz Akyildiz, Dan Crisan, Joaquin Miguez

TL;DR

This work introduces a sequential predictor-corrector Euler discretisation for multivariate Itô SDEs, achieving weak order $1$ and enabling blockwise updates that scale to high-dimensional state spaces. It provides a rigorous convergence analysis showing that discretised laws $\pi_k^h$ converge to the true posterior laws $\pi_k$ as $h\to 0$, and demonstrates this in a continuous-discrete Bayesian filtering setting. When integrated into ensemble Kalman filters, the sequential scheme yields improved robustness and efficiency, allowing larger time steps and smaller ensembles in the stochastic Lorenz 96 model. The results indicate practical gains for high-dimensional data assimilation tasks and point to fruitful extensions to RK schemes, SPDEs, MLMC, and particle filtering. Overall, the paper combines theoretical guarantees with empirical improvements, advancing time discretisation and filtering for complex stochastic systems.

Abstract

We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler-Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler-Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles and noisier systems.

Sequential discretisation schemes for a class of stochastic differential equations and their application to Bayesian filtering

TL;DR

This work introduces a sequential predictor-corrector Euler discretisation for multivariate Itô SDEs, achieving weak order and enabling blockwise updates that scale to high-dimensional state spaces. It provides a rigorous convergence analysis showing that discretised laws converge to the true posterior laws as , and demonstrates this in a continuous-discrete Bayesian filtering setting. When integrated into ensemble Kalman filters, the sequential scheme yields improved robustness and efficiency, allowing larger time steps and smaller ensembles in the stochastic Lorenz 96 model. The results indicate practical gains for high-dimensional data assimilation tasks and point to fruitful extensions to RK schemes, SPDEs, MLMC, and particle filtering. Overall, the paper combines theoretical guarantees with empirical improvements, advancing time discretisation and filtering for complex stochastic systems.

Abstract

We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler-Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler-Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles and noisier systems.
Paper Structure (36 sections, 3 theorems, 71 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 36 sections, 3 theorems, 71 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Discrete-time Bayesian filters
  4. Contributions
  5. A novel time-discretisation scheme for SDEs
  6. A convergence analysis for the approximate laws $\{\pi_k^h\}$
  7. A numerical study of the performance of the EnKF in the continuous-discrete-time framework
  8. Notation
  9. A sequential predictor-corrector numerical scheme
  10. The discretisation scheme
  11. Weak convergence
  12. Example: The stochastic Lorenz 96 model
  13. State-space models
  14. Observations over a time grid
  15. Continuous- and discrete-time Markov kernels
  16. ...and 21 more sections

Key Result

Theorem 2.4

\newlabelthWeak0 If Assumption asW holds, then, for any test function $\phi \in C_B^4\left( \mathbb{R}^{d_x} \right)$, where the constant $C={\mathcal{O}}(Td_x^2)<\infty$ is independent of the time step $h=\frac{T}{N}$ and the initial value $X(0)=X_0$.

Figures (4)

  • Figure 1: Sample trajectories with the standard Euler and sequential Euler methods. The variable plotted is $x_{100}(t)$ and the overall dimension is $d_x=200$. The diffusion factor is $\sigma=\sqrt{1/2}$. The horizontal axis is continuous time, $t \in [0,4]$. The time step, $h$, is indicated in the caption of each plot. For $h=10^{-2}$ and $h=5 \times 10^{-2}$ the standard Euler scheme does not complete the simulation.
  • Figure 1: Percentage of complete runs of the EnKF and SEnKF for varying time step $h$ (from top to bottom: $h=10^{-3}, 5 \times 10^{-3}$ and $10^{-2}$) and diffusion factor $\sigma$ (left: $\sigma=\sqrt{1/4}$, middle: $\sigma=\sqrt{1/2}$, right: $\sigma=1$). In each plot, the horizontal axis represents the variance ($\sigma_y^2$) of the observational noise. The length of the simulation is $T=5$ continuous-time units. The continuous-time gap between consecutive observations is $\Delta = 0.1$. The model dimension is $d_x=200$ and the size of the ensemble is $M=d_x=200$. The percentages are estimated from 300 independent simulations.
  • Figure 2: Performance of the standard Euler-Maruyama ('Euler') and the sequential predictor-corrector Euler ('seq. Euler') schemes. All graphs are averaged over 10,000 independent simulation runs. The length of the simulation interval is $T=2$ and the dimension of $X(t)$ is $d_x=200$. Left: Percentage of complete simulations. Middle: Normalised weak error versus the time step $h$. Right: Normalised weak error versus run-time in seconds.
  • Figure 2: Left: NMSE vs. ensemble size $M$ for the Euler EnKF, Euler SEnKF, sequential Euler EnKF and sequential Euler SEnKF algorithms in three different scenarios. Right: NMSE vs. run-time for the same set of simulations.

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.4
  • Proof 1
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.3
  • ...and 6 more